Multiattribute acceptance sampling plans - Library(ISI Kolkata ...

2.2 The expected cost model for discrete prior distributions
2.2.1 Assumptions
By Bayesian plans we understand the plans obtained by minimizing average costs which
has three identifiable components viz. inspection costs, acceptance costs, and rejection
costs. For these plans the process average defective is taken to be a random variable. In
our present context the prior distribution (i.e. the distribution of process average) is the
expected distribution of lot quality vector on which the sampling plan is going to operate.
For the multiattribute Bayesian plans considered by others [See chapter 2.1], the process
average for each attribute has been assumed to follow a beta distribution so that the lot
quality distribution for each attribute becomes a beta binomial.Thus, in a situation when
defect occurrences are jointly independent the product of individual beta distributions are
chosen as an appropriate prior.
We may, however, note that even when the process is in control with respect to such a
prior, the process will occasionally go out of control and some lots of poorer quality will be
produced before the process gets corrected. We then have a situation of a beta prior with
outliers.
As an alternative to these models, consider the process average vector as a random variable
which may take on two values,(p 1 , p 2 , ..., p r ) and (p ′ 1, p ′ 2, ..., p ′ r), a satisfactory and an
unsatisfactory quality level with given probabilities. This two point prior may be considered
as a simplification of the one point prior with outliers, because the model contains some
information about the distribution of the outlier. When the process performs at an unsatisfactory
level it generally happens (e.g. for a manufacturing operation) that the quality level
is poor for all the attributes. We cite one example from a real life situation.
2.2.2 A real life example
The example given below constitutes a typical example picked up from the author’s list
of similar applications in factories rendered by him as a QC professional. For the general
discussion intended here, the details relating to the particular application are not included
and only the relevant calculations are presented.
A company manufactures about 1.5 lakhs of 25 mm RS closures in a shift. A shift’s
production by a group is being packed in cartons, each carton is considered as an inspection
lot for verification after the end of the shift. The two important sets of attributes for the
product are functional defects and surface defects. On the spot observations have been made
using a p-chart data format on these attributes.
In a typical scenario one observes, that most of the time the process is stable at a certain
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